Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{t^3 + 15t^2 + 56t}{-3t^2 - 18t + 48}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {t(t^2 + 15t + 56)} {-3(t^2 + 6t - 16)} $ $ z = -\dfrac{t}{3} \cdot \dfrac{t^2 + 15t + 56}{t^2 + 6t - 16} $ Next factor the numerator and denominator. $ z = - \dfrac{t}{3} \cdot \dfrac{(t + 8)(t + 7)}{(t + 8)(t - 2)}$ Assuming $t \neq -8$ , we can cancel the $t + 8$ $ z = - \dfrac{t}{3} \cdot \dfrac{t + 7}{t - 2}$ Therefore: $ z = \dfrac{ -t(t + 7)}{ 3(t - 2)}$, $t \neq -8$